Symmetric monoidal category

In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" $$\otimes$$ is defined) such that the tensor product is symmetric (i.e. $$A\otimes B$$ is, in a certain strict sense, naturally isomorphic to $$B\otimes A$$ for all objects $$A$$ and $$B$$ of the category). One of the prototypical examples of a symmetric monoidal category is the category of vector spaces over some fixed field k, using the ordinary tensor product of vector spaces.

Definition
A symmetric monoidal category is a monoidal category (C, ⊗, I) such that, for every pair A, B of objects in C, there is an isomorphism $$s_{AB}: A \otimes B \to B \otimes A$$ called the swap map that is natural in both A and B and such that the following diagrams commute:
 * The unit coherence:
 * [[File:symmetric monoidal unit coherence.png]]
 * The associativity coherence:
 * [[File:symmetric monoidal associativity coherence.png]]
 * The inverse law:
 * [[File:symmetric monoidal inverse law.png]]

In the diagrams above, a, l, and r are the associativity isomorphism, the left unit isomorphism, and the right unit isomorphism respectively.

Examples
Some examples and non-examples of symmetric monoidal categories:
 * The category of sets. The tensor product is the set theoretic cartesian product, and any singleton can be fixed as the unit object.
 * The category of groups. Like before, the tensor product is just the cartesian product of groups, and the trivial group is the unit object.
 * More generally, any category with finite products, that is, a cartesian monoidal category, is symmetric monoidal. The tensor product is the direct product of objects, and any terminal object (empty product) is the unit object.
 * The category of bimodules over a ring R is monoidal (using the ordinary tensor product of modules), but not necessarily symmetric. If R is commutative, the category of left R-modules is symmetric monoidal. The latter example class includes the category of all vector spaces over a given field.
 * Given a field k and a group (or a Lie algebra over k), the category of all k-linear representations of the group (or of the Lie algebra) is a symmetric monoidal category. Here the standard tensor product of representations is used.
 * The categories (Ste,$$\circledast$$) and (Ste,$$\odot$$) of stereotype spaces over $${\mathbb C}$$ are symmetric monoidal, and moreover, (Ste,$$\circledast$$) is a closed symmetric monoidal category with the internal hom-functor $$\oslash$$.

Properties
The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an $$E_\infty$$ space, so its group completion is an infinite loop space.

Specializations
A dagger symmetric monoidal category is a symmetric monoidal category with a compatible dagger structure.

A cosmos is a complete cocomplete closed symmetric monoidal category.

Generalizations
In a symmetric monoidal category, the natural isomorphisms $$s_{AB}: A \otimes B \to B \otimes A$$ are their own inverses in the sense that $$s_{BA}\circ s_{AB}=1_{A\otimes B}$$. If we abandon this requirement (but still require that $$A\otimes B$$ be naturally isomorphic to $$B\otimes A$$), we obtain the more general notion of a braided monoidal category.